Question

这个不定积分怎么算？

求过程

Answer 1

I = ∫x(secx)^4dx = ∫x(secx)^2dtanx = ∫x[(tanx)^2+1]dtanx
= (1/3)∫xd(tanx)^3 + ∫xdtanx
= (1/3)x(tanx)^3 - ∫(tanx)^3dx + xtanx - ∫tanxdx
= (1/3)x(tanx)^3 - ∫(tanx)^3dx + xtanx + ln|cosx|
其中 I1 = ∫(tanx)^3dx = ∫(sinx)^3dx/(cosx)^3
= -∫[1-(cosx)^2]dcosx/(cosx)^3
= -∫dcosx/(cosx)^3 + ∫dcosx/cosx
= 1/[2(cosx)^2] + ln|cosx|
则 I = (1/3)x(tanx)^3 - 1/[2(cosx)^2] + xtanx + C

Answer 2

令x=tant，则dx=sec^2tdt
原式=∫(tant*e^t)/sec^3t*sec^2tdt
=∫sint*e^tdt
=∫sint*d(e^t)
=sint*e^t-∫e^t*costdt
=sint*e^t-∫cost*d(e^t)
=sint*e^t-cost*e^t-∫e^t*sintdt
即∫sint*e^tdt=e^t*(sint-cost)/2+C
原式=e^(arctanx)*[(x-1)/√(1+x^2)]+C，其中C是任意常数